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Existence of embedded minimal disks

Baris Coskunuzer

UT Dallas, Department of Mathematical Sciences, Richardson, TX 75080, USA

E-mail Address: baris.coskunuzer@utdallas.edu

https://doi.org/10.1142/S0219199721500504Cited by:0 (Source: Crossref)

Abstract

We give a generalization of Meeks–Yau’s celebrated embeddedness result for the solutions of the Plateau problem for extreme curves.

Keywords:

AMSC: 53A10, 53C42

References

[1] F. J. Almgren and L. Simon, Existence of embedded solutions of Plateau’s problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 6(3) (1979) 447–495. Google Scholar
[2] F. J. Almgren and W. P. Thurston, Examples of unknotted curves which bound only surfaces of high genus within their convex hulls, Ann. Math. 105 (1977) 527–538. Crossref, Web of Science, Google Scholar
[3] B. Coskunuzer, Examples of area minimizing surfaces in 3-manifolds, Int. Math. Res. Not. 6 (2014) 1613–1634. Crossref, Web of Science, Google Scholar
[4] B. Coskunuzer, Embedded Plateau problem, Trans. Amer. Math. Soc. 364 (2012) 1211–1224. Crossref, Web of Science, Google Scholar
[5] J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931) 263–321. Crossref, Google Scholar
[6] T. Ekholm, B. White and D. Wienholtz, Embeddedness of minimal surfaces with total boundary curvature at most $4 π$ $4 π$ , Ann. Math. 155(1) (2002) 209–234. Crossref, Web of Science, Google Scholar
[7] H. Federer, Geometric Measure Theory (Springer-Verlag, New York, 1969). Google Scholar
[8] R. Gulliver and J. Spruck, On embedded minimal surfaces, Ann. Math. 103 (1976) 331–347. Crossref, Web of Science, Google Scholar
[9] J. Hass and P. Scott, The existence of least area surfaces in $3$ $3$ -manifolds, Trans. Amer. Math. Soc. 310 (1988) 87–114. Crossref, Web of Science, Google Scholar
[10] R. Hardt and L. Simon, Boundary regularity and embedded solutions for the oriented Plateau problem, Ann. Math. 110 (1979) 439–486. Crossref, Web of Science, Google Scholar
[11] J. H. Hubbard, On the convex hull genus of space curves, Topology 79 (1980) 203–208. Crossref, Google Scholar
[12] C. B. Morrey, The problem of Plateau on a Riemannian manifold, Ann. Math. 49 (1948) 807–851. Crossref, Web of Science, Google Scholar
[13] W. Meeks and S. T. Yau, Topology of three-dimensional manifolds and the embedding problems in minimal surface theory, Ann. Math. 112 (1980) 441–484. Crossref, Web of Science, Google Scholar
[14] W. Meeks and S. T. Yau, The classical Plateau problem and the topology of three manifolds, Topology 21 (1982) 409–442. Crossref, Google Scholar
[15] W. Meeks and S. T. Yau, The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z. 179 (1982) 151–168. Crossref, Web of Science, Google Scholar
[16] T. Rado, On Plateau’s problem, Ann. Math. 31 (1930) 457–469. Crossref, Google Scholar
[17] F. Tomi and A. J. Tromba, Extreme curves bound embedded minimal surfaces of the type of the disc, Math. Z. 158 (1978) 137–145. Crossref, Web of Science, Google Scholar
[18] A. J. Tromba, The Set of Curves of Uniqueness for Plateau’s Problem has a Dense Interior, Lecture Notes in Mathematics, Vol. 597 (Springer, Berlin, 1977), pp. 696–706. Crossref, Google Scholar
[19] B. White, Existence of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on $3$ $3$ -manifolds, J. Diff. Geom. 33 (1991) 413–443. Web of Science, Google Scholar